All-energy stability of the largest mode in the convex case

Let $m,n$ be the mode indices, let $P$ be the parameter, and let $\Theta_m$ and $\Theta_n$ denote the corresponding modes with energies $E_{\Theta_m}$ and $E_{\Theta_n}$. Assume

$$0\leqslant P\leqslant n^2<m^2.$$

All-energy stability conjecture. Under these assumptions, $\Theta_m$ is linearly stable with respect to $\Theta_n$ for every $E_{\Theta_m}>0$.

The preceding theorem proves stability for sufficiently small and sufficiently large energies, while the conjecture asserts that no intermediate-energy instability occurs.